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We develop a projected least squares estimator for the change point parameterin a high dimensional time series model with a potential change point.Importantly we work under the setup where the jump size may be near theboundary of the region of detectability. The proposed methodology yields anoptimal rate of convergence despite high dimensionality of the assumed modeland a potentially diminishing jump size. The limiting distribution of thisestimate is derived, thereby allowing construction of a confidence interval forthe location of the change point. A secondary near optimal estimate is proposedwhich is required for the implementation of the optimal projected least squaresestimate. The prestep estimation procedure is designed to also agnosticallydetect the case where no change point exists, thereby removing the need topretest for the existence of a change point for the implementation of theinference methodology. Our results are presented under a general positivedefinite spatial dependence setup, assuming no special structure on thisdependence. The proposed methodology is designed to be highly scalable, andapplicable to very large data. Theoretical results regarding detection andestimation consistency and the limiting distribution are numerically supportedvia monte carlo simulations.

Inference on the change point with the jump size near the boundary of the region of detectability in high dimensional time series models